Toward a Generalization of the Gross-Zagier Conjecture

by William Stein

March 2009

To Appear in IMRN

Download it now as a PDF
International Mathematics Research Notices 2010; doi: 10.1093/imrn/rnq075 (free download at this page)


We review some of Kolyvagin's results and conjectures about elliptic curves, then make a new conjecture that slightly refines Kolyvagin's conjectures. We introduce a definition of finite index subgroups Wp of E(K), one for each prime p that is inert in a fixed imaginary quadratic field K. These subgroups generalize the group Z yK generated by the Heegner point yK in E(K) in the case ran = 1. For any curve with ran ≥ 1, we give a description of Wp, which is conditional on truth of the Birch and Swinnerton-Dyer conjecture and our conjectural refinement of Kolyvagin's conjecture. We then deduce the following conditional theorem, up to an explicit finite set of primes: (a) the set of indexes [E(K): Wp] is finite, and (b) the subgroups Wp with [E(K): Wp] maximal are exactly the subgroups that satisfy a higher-rank generalization of the Gross-Zagier formula. We also investigate a higher-rank generalization of a conjecture of Gross-Zagier.

NOTE: The main reference for this paper is Kolyvagin's paper "On the Structure of Selmer groups". I have LaTeX'd that paper here to make it easier for you to refer to.